Theorem eexinst11

Description: exinst11 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eexinst11.1	$$⊢\phi \to \exists x\psi$$
	eexinst11.2	$$⊢\phi \to \left(\psi \to \chi \right)$$
	eexinst11.3	$$⊢\phi \to \forall x\phi$$
	eexinst11.4	$$⊢\chi \to \forall x\chi$$
Assertion	eexinst11	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		eexinst11.1	$$⊢\phi \to \exists x\psi$$
2		eexinst11.2	$$⊢\phi \to \left(\psi \to \chi \right)$$
3		eexinst11.3	$$⊢\phi \to \forall x\phi$$
4		eexinst11.4	$$⊢\chi \to \forall x\chi$$
5	3 4 2	exlimdh	$$⊢\phi \to \left(\exists x\psi \to \chi \right)$$
6	1 5	syl5com	$$⊢\phi \to \left(\phi \to \chi \right)$$
7	6	pm2.43i	$$⊢\phi \to \chi$$